Chemically-operated turing machine

ABSTRACT

The present disclosure relates to a Turing machine having a reactor comprising a reactant solution comprising a reactant; a first chemical species source to provide a selected amount of a first chemical species; a second chemical species source to provide a selected amount of a second chemical species; one or more controllers coupled to control the addition of the first and second chemical species from the first and second chemical species sources responsive to an input; and a sensor positioned to sense changes in the reactant as the controller controls the first and second chemical species sources to add selected amounts of the respective first and second chemical species to the reactor. The controller receives signals corresponding to the state of the reactant and correlates the states of the reactant to a result that is computed as a function of the input.

FIELD OF INVENTION

The present disclosure relates to a chemically-operated Turing machine.

BACKGROUND

The universal Turing machine was devised in 1936 by Alan Turing. It was intended to mimic the pencil-and-paper operations of a mathematician. A Turing machine is a model of computation, or a way of representing and performing a given computation by means of some algorithm which is also known as the machine's “procedure.” Turing machines are mathematically and logically equivalent to many other models of computation, such as cellular automata, neural networks, and digital computers. Because no model of computation is more powerful than a Turing machine, it is considered to embody what is meant when a problem is referred to as being “computable”. In other words, anything for which an algorithm can be written, can be computed by a Turing machine. Turing machines have facilitated the proof of many important ideas and theorems regarding the nature and limits of computation, such as the undecidability of the halting problem and the existence of uncomputable functions.

While a Turing machine may be constructed to implement any specific algorithm imaginable, it is impractical to build a physical machine to solve each new problem. Fortunately, Turing machines can be constructed that take as an input a description and data tape from another Turing machine, and simulate that Turing machine on its own tape. Such a Turing machine is known as a Universal Turing Machine (UTM). Personal computers are good approximations of Universal Turing Machines, in that the programs that they run are descriptions of specific algorithms and hence, specific Turing machines. Personal computers fall short of UTMs, however, because their memory cannot be expanded every time more storage is needed. There is therefore a need in the art for Turing machines that overcome the shortcomings of, e.g., personal computers, and more closely approximate a UTM.

SUMMARY

A first aspect of the invention is a Turing machine based on an oscillatory chemical reaction which comprises a reactor comprising a reactant solution comprising a reactant; a first chemical species source to provide a selected amount of a first chemical species; a second chemical species source to provide a selected amount of a second chemical species; one or more controllers coupled to control the addition of the first and second chemical species from the first and second chemical species sources responsive to an input; and one or more sensors positioned to sense changes in the reactant as the controller controls the first and second chemical species sources to add selected amounts of the respective first and second chemical species to the reactor, wherein the controller receives signals corresponding to the state of the reactant and correlates the states of the reactant to a result that is computed as a function of the input.

A second aspect of the invention is the Turing machine as defined in the first aspect further comprising a tape to provide the input to the controller.

A third aspect of the invention is the Turing machine as defined in any of the previous aspects, wherein the one or more sensors of changes in reactant comprise a redox sensor, a pH sensor, a temperature sensor, a pressure sensor, a UV-Vis sensor or combinations thereof.

A fourth aspect of the invention is the Turing machine as defined in any of the previous aspects, wherein the first chemical species comprises an oxidizing agent and the second chemical species comprises a reducing agent.

A fifth aspect of the invention is the Turing machine of the fourth aspect wherein the oxidizing agent is bromate ions.

A sixth aspect of the invention is the Turing machine as defined in the fourth aspect of the invention, wherein the reducing agent is malonic acid.

A seventh aspect of the invention is the Turing machine as defined in any of the previous aspects, wherein the reactor is a continuously stirred tank reactor.

An eighth aspect of the invention is the Turing machine as defined in any of the previous aspects, wherein the sensor comprises a spectrometer adapted to periodically detect color changes in the reactant solution.

A ninth aspect of the invention is the Turing machine as defined in any of the previous aspects, wherein the first chemical species source and/or the second chemical species source comprise burettes or syringe pumps.

A tenth aspect of the invention is the Turing machine as defined in any of the previous aspects, wherein the reactant comprises a compound capable of attaining meta stable states or an oscillatory regime.

A eleventh aspect of the invention is the Turing machine as defined in any of the previous aspects, wherein the reactant comprises a transition metal complex.

A twelfth aspect of the invention is the Turing machine as defined in any of the previous aspects, wherein the transition metal complex is a ruthenium complex, a cerium complex, an iron complex or a cobalt complex.

A thirteenth aspect of the invention is the Turing machine as defined in the twelfth aspect of the invention, wherein the ruthenium complex is a tris(bipyridine)ruthenium (II) complex.

A fourteenth aspect of the invention is the Turing machine as defined in any of the previous aspects, wherein, the input of the Turing machine as defined above comprises a parenthesis or a string of parentheses.

A fifteenth aspect of the invention is the Turing machine as defined in any of the previous aspects, wherein said Turing machine is a parenthesis checker.

A sixteenth aspect of the invention is the use of the Turing Machine, as defined in any of the previous aspects, as an element of the central processing unit of a programmable chemical computer.

A seventeenth aspect of the invention is a central processing unit of a programmable chemical computer comprising one or more Turing Machines, as defined in any of the previous aspects, or appropriate variants thereof.

An eighteenth aspect of the invention is a programmable chemical computer comprising a central processing unit which comprises one or more Turing Machines, as defined in any of the previous aspects, or appropriate variants thereof.

A nineteenth aspect of the invention is a method of operating a chemical Turing machine based on an oscillatory chemical reaction comprising a reactor comprising a reactant solution comprising a reactant; providing an input to a controller, coupled to control the addition of a first and a second chemical species from a first chemical species source and a second chemical species source, responsive to the input; and sensing changes in the reactor as the controller controls the first and second chemical species sources to add selected amounts of the respective first and second chemical species to the reactor, wherein the controller receives signals corresponding to the states of the reactant; and correlating the states of the reactant to a result that is computed as a function of the input.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic representation of a finite state machine (FSM) according to an example embodiment.

FIGS. 2 and 3 are nomographs (i.e., pre-calibrated graphs against which one compares the results of the computation taking place and the result being interpreted).

FIG. 4 is a schematic representation of an FSM (Finite State Machine) according to an example embodiment.

FIGS. 5A and 5B are standard schematic logical representations of a chemically-operated Turing machine of the embodiments of the present invention, including an input tape and head; a logic counter tape and head; a locator tape and head; and output tape and head.

FIGS. 6 and 7 are nomographs.

DETAILED DESCRIPTION

The disclosure presents the design, realization, and operation of embodiments of a chemically-operated Turing machine. In some embodiments, the chemically-operated Turing machine combines a potentially infinite input tape and a finite state machine (FSM) that uses chemical reactions. In some embodiments, once the information in the input tape has been fed into the chemically-operated Turing machine of the embodiments of the present invention, the chemically-operated Turing machine of the embodiments of the present invention uses chemical energy (i.e., chemical reactions) for all the features involved in its operation, including its logical state transitions, and does not require any mechanical, electrical, electronic or any other form of external intervention during the course of its logical operation.

In some embodiments, the finite state machine is based on the dynamics of the states of the well-known and extensively documented properties of the multistate semi-batch or batch (e.g., intermittent flow) or continuously-stirred tank reactor (CSTR) version of the Belousov/Zhabotinsky (B/Z) reaction. In this embodiment, the “alphabet” (e.g., a two-member alphabet or a two-letter alphabet) on which information is entered into the machine through the input tape is based on a two-letter or a two-symbol alphabet since it corresponds to the addition of two different chemical species. Depending on the specific order in which pre-determined amounts of two different chemical species are added to a reactor comprising a reactant solution comprising a reactant (e.g., in an aqueous solution), the B/Z reaction is driven into a finite number of discrete states. These states are then sensed/detected and can be interpreted as the result of the “computation” performed by the chemically-operated Turing machine in response to the particular ordered sequence of stimuli/inputs provided by the input tape. The energy source for the operation of the chemically-operated Turing machine is the chemical energy expended in the chemical reactions taking place during the computations.

In some embodiments, the “alphabet” on which information is entered in the input tape of the chemically-operated Turing machine described herein comprises an open parenthesis “(“and a closed parenthesis ”).” In such embodiments, a chemically-operated Turing machine as described here can be configured as the basic “parenthesis checker” to decide whether a sequence of open and closed parentheses is well-formed, i.e., if for every open parenthesis in the sequence there exists a corresponding closed parenthesis. In fact, the chemically-operated Turing machine is, in some embodiments, the chemical generalization of the generic canonical parenthesis checker constructed by Minsky in the 1950's. See, e.g., Minsky 1967, Computation: Finite and Infinite

Machines, Prentice Hall. Minsky's machine is well known to be one of the simplest Turing machines. To date such machines have been constructed based on mechanics, electronics and, theoretically, using enzymes with known chemical properties, but their actual physical or chemical implementation from first principles has not been realized. Machines like Minsky's have never been designed and implemented using chemical means; that is, without making use of, or making any reference to, the chemistry of extant living systems and with full control by the designer over its functionality, operation, design and construction.

The chemically-operated Turing machine of the embodiments of the present invention is the practical realization of the concept of a chemical computation carried out in any inorganic (e.g., independent from extant life) system, and is based on an oscillatory chemical reaction. Its principles are general and can be extended, for example, to any inorganic-chemical system or any organometallic chemical system. The chemically-operated Turing machine of the embodiments of the present invention opens the way for the construction of chemically-based computers that are chemically programmed and chemically operated and have the ability to execute preprogrammed functions, carry out operations, and handle information in a completely autonomous manner. Such chemically-based computers may be capable of chemically responding to chemical stimuli in such a way that the result is an ordered chemical response, uniquely related to the stimuli on an input tape, that is the result of the implementation via chemistry of some effective procedure (i.e., an algorithm).

The chemically operated Turing Machine of the embodiments of the present invention is, however, not restricted to inorganic chemistry. It is a general construct, and as such it can also be implemented in biochemical systems, both naturally occurring and synthetic. This is so, as long as the biochemical system is oscillatory and has, at least, two substrates or one substrate and an additional substance that can behave like a substrate for the purposes of the oscillation. The oscillatory nature of the chemistry is needed in order to deal with the presence of non-linearities in the procedure (or algorithm) implemented by the Turing Machine of some embodiments of the present invention. In non-chemical implementations of Turing machines this is done via ON-OFF switches, such as relays which, appropriately interconnected, provide for the expansion of the non-linearities in the procedure as a power-series expansion (Shannon 1940, A Symbolic Analysis of Relays and Switching Circuits, M. Sc. Thesis, MIT; Lloyd 1992, Phys. Lett. A 167 255-260. In the chemically operated Turing Machine of various embodiments of the present invention, this is done instead by an expansion of the non-linearity using a Fourier series. The oscillations in the chemistry provide the elements for the sine and cosine functions which are the basis for the expansion in a Fourier series. The need for at least two substrates (or one substrate and an additional substance that can behave like a substrate for the purposes of the oscillation) for the oscillatory reaction appears because of the necessity to provide the “letters” for the alphabet in which the problem (for example an arbitrary sequence of open and close parentheses) is submitted to the chemically operated Turing Machine of the various embodiments of the present invention. Indeed a two-letter alphabet is the minimal length of an alphabet needed to express, perhaps in a codified version, any message of arbitrary length.

Therefore, the chemically operated Turing Machine of various embodiments of the present invention is independent of the nature or origin or specific chemistry. Both inorganic and organic and, in either case, synthetic or natural will operate the Machine. So long as one uses an oscillatory chemical reaction where there are two substrates (or one substrate and an additional substance that can behave like a substrate for the purposes of the oscillation) a realization is possible for any person with knowledge of the art. The two substrate oscillator can be based on inorganic chemistry (as in the case of the Belousov-Zhabotinsky reaction) in synthetic organic oscillators or bio-oscillators (as in the case of synthetic networks of transcriptional regulators (Elowitz and Liebler 2000, Nature 403:335-338) or in organic chemistry and extant natural biochemistry (as in the case of glycolytic oscillations (Sel'kov 1968, European J. Biochem. 4:79-86; Hess and Boiteaux 1971, Annu Rev Biochem. 40:237-258; Chance, Pye, Ghosh and Hess 1973, Biological and Biochemical Oscillators (Academic Press); Novak and Tyson 2008, Nat Rev Mol Cell Biol. 12:981-991).

One example of a “bio-oscillator” is provided by the photoluminescent enzyme phosphofructokinase (PFK), e.g. Tyson 2002, Biochemical Oscillations, in Fall, Marland, Wagner, Tyson, eds., Computational Cell Biology (Springer-Verlag) 230-260, which phosphorylates fructose-6-phosphate to produce fructose-1,6-biphosphate with adenosine triphosphate (ATP) as the phosphate donor. In glycolysis ATP is both a substrate and an inhibitor of PFK, while the product of the reaction, adenosine diphosphate (ADP), is also an activator of the enzyme. ADP would be analogous to the first chemical species, while ATP would be analogous to the second chemical species in the chemically operated Turing Machine of the embodiments of this invention. In addition, instead of monitoring the oscillations of the RedOx potential of the reactant solution as is done in the Belousov-Zhabotinsky inorganic embodiment, here one monitors the fluorescence of the enzyme PFK and its subsequent oscillations and changes in amplitude as a sequence of drops of ADP and ATP are fed to the Turing Machine.

The chemically-operated Turing Machine of the present invention is based on an oscillatory chemical reaction and comprises:

a reactor comprising a reactant solution comprising a reactant;

a first chemical species source to provide a selected amount of a first chemical species;

a second chemical species source to provide a selected amount of a second chemical species;

one or more controllers coupled to control the addition of the first and second chemical species from the first and second chemical species sources responsive to an input; and

one or more sensors positioned to sense changes in the reactant as the controller controls the first and second chemical species sources to add selected amounts of the respective first and second chemical species to the reactor, wherein the controller receives signals corresponding to the state of the reactant and correlates the states of the reactant to a result that is computed as a function of the input.

As used herein, the term “reactant solution” includes, but is not limited to a reactant that is dissolved in a solvent. The solvent can be any suitable solvent or combination of solvents. Said solvent can be selected from water, C₁-C₄ alcohols such as methanol, ethanol, iso-propanol, t-butanol, and the like and mixtures thereof. Cosolvents like for instance dimethylformamide and dimethylsulfoxide may also be present in the reactant solution. Finally, acids (e.g., aqueous acid solutions comprising sulfuric acid, nitric acid or any other equivalent strong inorganic acid or mixtures thereof) can also be present in the reactant solution.

In some embodiments, the reactor comprising the reactant solution may be a semi-batch or batch (e.g., intermittent flow) or a continuously stirred tank reactor (CSTR) filled with the reactant solution. The reactor can be of any suitable size from picoliter size, to nanoliter size, to microliter size to multi-liter size to pilot-plant scale, and even industrial-scale. Those of skill in the art will be able to determine the appropriate size of the reactor.

In some embodiments, the reactant solution is acidic. The reactant solution may be acidified using acid solutions (e.g., aqueous acid solutions) comprising sulfuric acid, nitric acid or any other equivalent strong inorganic acid or mixtures thereof.

As used herein, the term “reactant” includes, but is not limited to, a compound capable of attaining meta stable states or an oscillatory regime, which can also be referred to as an oscillatory system that has two substrates or one substrate and an additional substance that can behave like a substrate for the purposes of oscillation. Said compound or oscillatory system can be inorganic, organometallic or organic in nature and, in either case, synthetic or natural.

As used herein, the term “meta stable states” broadly refers to an unstable and transient, but relatively long-lived state of a chemical system.

Compounds having meta stable states or oscillatory systems as referred above include, but are not limited to, transition metal complex/catalysts as, for instance, ruthenium (II) complexes, ruthenium (III) complexes, cerium (III) complexes or cerium (IV) complexes, where the transition metal complex/catalyst would oscillate between two oxidation states (e.g., ruthenium (II) and ruthernium (III)). In some embodiments, the ligands of the transition metal complex are bipyridine as, for instance, in tris(bipyridine)ruthenium (II) complex. In a general definition, oscillatory systems can be formed by, for instance, ruthenium (II)/ruthenium (III), cerium (III)/cerium (IV) or Fe (II)/Fe (III). Other possible oscillatory system according to the present invention are based on the pair I₂/IO₃ ⁻. This system works in the presence of hydrogen peroxide (Bray 1921, JACS. 43 (6):1262; Liebhafsky 1969, Anal. Chem., 4:1894-1897) and, additionally, in the presence of manganese (II) (Briggs & Rauscher 1973 J. Chem. Ed. 50: 496). Further oscillatory systems according to the present invention employ sulfur, phosphorous or cobalt; an example of bioscillator is provided by the phosphofructokinase (PFK) system in glycolysis, Hess and Boiteaux 1973 op. cit. In a particular embodiment, the transition metal complex is ferroin-ferriin or other transition metal ions or complexes that possess at least two oxidation states differing in a single electron and that change the color of the solution when changing from one oxidation state to the other. Mixtures of any of these catalysts are also contemplated.

It should now be clear that all that the Turing Machine of various embodiments of the present invention requires to operate is an oscillatory reaction. Different oscillatory reactions suitable for the present invention are described in the literature. For example, Noyes and Field 1974, Journal of Chemical Physics, 60(5):1877-1884 and Epstein et al. 2003, Dalton Trans. 1201-1217, included herein by reference, provide a review of known oscillatory reactions and general rules for the systematic search of new ones by combining a generic mathematical model, a continuous flow stirred tank reactor to keep the system away from equilibrium, and inorganic reaction kinetics. This approach described by Epstein et al. 2003, Dalton Trans. 1201-1217 is based on the following principles:

(1) Sustained oscillation can occur if a system is kept away from equilibrium. One way to do this is to run the reaction in a flow reactor, which allows for continuous input of fresh reactants and outflow of products.

(2) Autocatalytic reactions sometimes exhibit bistable behavior when run in a flow reactor. That is, for certain sets of input concentrations and flow rate, the system may, depending upon its history, reach either of two steady states, each of which is stable to small perturbations.

(3) If a bistable system is subjected to a feedback that affects the concentration of the autocatalytic species on a time scale long with respect to the characteristic times for the system to relax to its steady states, then by intensifying the feedback, it should be possible to cause the system to oscillate, essentially between the two, no longer stable, steady states.

(4) The situation described above can be generated by choosing an autocatalytic reaction, running it in a flow reactor to determine conditions for bistability, and then adding a feedback species that reacts sufficiently slowly with the appropriate species in the autocatalytic reaction. Increasing the concentration of the feedback species in the input flow should bring the system into its oscillatory state.

Thus, the Turing Machine of the invention may function with well known oscillating reactions, such as the above-mentioned Belousov-Zhabotinsky reaction, the Briggs-Rauscher reaction or the Bray-Liebhafsky reaction, and also with more recent systems discovered following the above mentioned approach or other means. Further, although not the only ones possible, systems that may be appropriate are based on sulfur (Orbán and Epstein 1985, J. Am. Chem. Soc., 107: 2302-2305; Fredrichs, Mlnarik, Grun and Thompson 2001, J. Phys. Chem. A, 105: 829-837), phosphorus (K. Kurin-Csörgei, M. Orbán, A. M. Zhabotinsky and I. R. Epstein, 2001, Faraday Discuss., 120: 11-19), cobalt (He, Kustin, Nagypál and Peintler 1994, Inorg. Chem., 33: 2077-2078) and manganese chemistry (Doona, Kustin, Orbán and Epstein 1991, J. Am. Chem. Soc., 113: 7484-7489) as well as organic reactions (J. H. Jensen 1983, J. Am. Chem. Soc., 105: 2639-2641). They also give rise to oscillating reactions with an average amplitude of oscillations value, ρ, and a frequency of oscillation ƒ that can be measured and used to perform a chemical computation in the Turing Machine of the present invention.

In some embodiments, the reactor comprising the reactant solution comprising the reactant may be equipped with a temperature controller. In some embodiments the temperature controller can maintain the temperature of the reactor, which is preferably between 15 to 25° C., more preferably from 20 to 25° C., to within ±0.2° C., such that the B/Z reaction may be carried out close to isothermal conditions. In other embodiments, the reactor comprising the reactant solution may be equipped with a reduction-oxidation (red-ox) meter to monitor the red-ox potential within the reactor. In still other embodiments, the reactor comprising the reactant solution comprising the reactant is equipped with a spectrophotometer (e.g., a UV-Vis spectrophotometer) to determine changes in absorbance and absorbance intensity of the solution in the reactor. In some embodiments, absorbance measurements can be taken every 100 milliseconds, selected so that one can monitor the evolution of the chemical reactions as they approach their metastable states. This process can take several periods of oscillation of the B/Z reaction operating under certain conditions and therefore in about 100 seconds, about 1000 absorbance measurements can be taken.

In some embodiments, the first chemical species source can be a burette, a drop counter, a syringe pump or any means that could contain the first chemical species (e.g., a vesicle). In some embodiments, the first chemical species source can be a burette. In the case of a burette, the burette is equipped with a controller coupled to control the addition of the first chemical species from the first chemical species source in response to an input. In the context of a burette, the controller may be, in some embodiments, a stopcock. In some embodiments, the controller can also be a syringe pump, a solenoid valve, microfluidic or chemically operated gel valves, or the like. In other embodiments, the controller may be any means by which the first chemical species can be released, including vesicles containing the first chemical species that would release (e.g., by rupturing) the first chemical species in response to an input.

The first chemical species source allows for the addition of discrete amounts of the first chemical species to the reactor comprising the reactant solution comprising the reactant. In some embodiments, the first chemical species comprises an oxidizing agent. In some embodiments, said oxidizing agent is in the form of a solid, preferably in the form of crystals; alternatively, the oxidizing agent is in the form of a solution.

As used herein, the term “oxidizing agent solution” includes, but is not limited to, an oxidizing agent that is dissolved in a solvent. Said solvent can be selected from water, C₁-C₄ alcohols and mixtures thereof. Oxidizing agents include, but are not limited to, bromate (⁻BrO₃) ions, iodate (⁻IO₃) ions, and the like. Oxidizing agents comprising bromate ions include, but are not limited to, lithium bromate, potassium bromate, sodium bromate, or any other soluble bromate salts of alkali metals and mixtures thereof.

In some embodiments, the second chemical species source can be a burette, a drop counter, a syringe pump or any means that could contain the second chemical species (e.g., a vesicle). In some embodiments, the second chemical species source can be a burette. In the case of a burette, the burette is equipped with a controller coupled to control the addition of the second chemical species from the second chemical species source in response to an input. In the context of a burette, the controller may be, in some embodiments, a stopcock. In some embodiments, the controller can also be a syringe pump, a solenoid valve, microfluidic or chemically operated gel valves, or the like. In other embodiments, the controller may be any means by which the second chemical species can be released, including vesicles containing the second chemical species that would release (e.g., by rupturing) the second chemical species in response to an input.

The second chemical species source allows for the addition of discrete amounts of the second chemical species to the reactor comprising the reactant solution comprising the reactant. In some embodiments, the second chemical species comprises a reducing agent. In some embodiments, said reducing agent is in the form of a solid, preferably in the form of crystals; alternatively, the reducing agent is in the form of a solution.

As used herein, the term “reducing agent solution” includes, but is not limited to, a reducing agent that is dissolved in a solvent. Said solvent can be selected from water, C₁-C₄ alcohols and mixtures thereof. Reducing agents include, but are not limited to, malonic acid, ascorbic acid, carbonic acid, citric acid, succinic acid or other suitable dicarboxylic acids, ketones or diketones, and mixtures thereof.

The one or more sensors can be any type of sensor known in the art. In some embodiments, the one or more sensors can be a redox sensor, a pH sensor, a temperature sensor, a pressure sensor, a UV-Vis sensor or combinations thereof.

Those of skill in the art will recognize that there are many variants of the B/Z reaction, using different types of oxidizing agents, reducing agents and metal complexes. Hence it is contemplated that one can adapt the machine to handle different strings by building equivalent Turing Machines with variants of the B/Z reaction.

Those of skill in the art will also recognize that changing the reactant solution comprising the reactant (e.g., by changing the reactant, including changing the transition metal complex/catalyst), the first chemical species, and/or the second chemical species may result in changes to the kinetic rates of one or several of the individual reactions in the B/Z reaction, changing, in turn, the oscillation characteristics. Acccordingly, B/Z reaction variants can be explored in order to design a chemically-operated Turing Machine with the desired response adapted to the available or desired monitoring possibilities.

The initial concentration of the reactant in the reactant solution is that which can be set into an oscillatory mode by the addition of second chemical species in the presence of the corresponding amounts of the first chemical species, and drives the reaction into an oscillatory mode. The establishment of the concentration of reactant and eventual acid in the reactant solution, as well as the concentrations of the oxiding and reducing agents in the chemical species sources and the respective amounts (drop volume) of the oxidizing and reducing agents to be added to the reagent solution will be routine work for those skilled in the art (Noyes and Furrow 1982). However, more specifically, the initial concentration of the reactant in the reactant solution, for instance of a ruthenium or cerium complex, is preferably between 10⁻⁵ M to 10⁻² M, more preferably between 10⁻⁴ M to 10⁻³ M. The initial concentration of acid in the reagent solution is preferably between 10⁻² M and 1 M, more preferably between 10⁻¹ M and 1 M. Further, the concentration of the oxidizing agent in the oxidizing agent solution is preferably between 5 M to 20 M, more preferably between 10 M to 15 M; whereas the concentration of the reducing agent in the reducing agent solution is preferably between 1 M to 15 M, more preferably between 5M to 10 M. Finally, the drop volume for the oxiding and reducing solutions is preferably between 1.0 to 0.4 mL, preferably from 0.8 to 0.5 mL.

As discussed in greater detail below, a chemically-operated Turing machine can also be achieved by changing the relative concentration of the first and second chemical species used to build an input tape, to the point where the solid form of the first and second chemical species, without dilution in a solvent, could be used. For example, crystals of the first chemical species and crystals of the second chemical species, adjusting for stoichiometry, may be used instead of a solution of the first chemical species and a solution of the second chemical species.

Even the definition or chemical identification of the input alphabet can be changed (e.g., bromate ions representing a closed parenthesis and malonic acid representing an open parenthesis). For any such chemically-operated Turing machines one could associate a nomograph (i.e., a pre-calibrated graph against which one compares the results of the computation taking place and the result being interpreted; see Examples) in which responses lying at any stage of computation above the nomograph are illogical.

The chemically-operated Turing Machine of the embodiments of the present invention comprises a finite state machine (FSM) and an input tape. The FSM has, in some embodiments, five states plus an initial state and a final state. These five states are described in greater detail in the Examples provided herein. The FSM, however, can have more than five states plus an initial state and a final state.

In some embodiments, the logical operation of the machine can be described by a set of four tapes and heads which, although not physical, help to capture and represent the specific features of the chemistry. The FSM is implemented as a reactor comprising a reactant that is ready to go into multiple states as soon as certain chemicals (i.e., the first and second chemical species) are added to the reactor. The computations take place in the FSM upon introduction of a sequence of stimuli in the form of drops of either of two substances (i.e., the first chemical species and the second chemical species) corresponding to the two letters of a two symbol alphabet, viz., “(“and”)”.

The sequence is contained in the input tape. When added to the reactant in the FSM, the chemistry has been designed in such a way that these stimuli to chemical reactions produce chemical results which, in turn, act as stimuli for subsequent states in the FSM. The results are the chemical result of the ensuing activity of the chemical reactions that constitute the five (plus initial) states of the FSM. These include oscillatory states, and the results are manifested by changes in the frequency of oscillation between colors and in the average intensity of their hues as they manifest in the reactant solution. These changes can be readily appreciated by the unaided eye or, in some embodiments, with a spectrophotometer.

In some embodiments, the FSM can take the form of a reactor 100 comprising a reactant solution 102, as shown in FIG. 1. See also, FIG. 4. A first chemical species source 106, in this case a first burette, is used to provide a selected amount of the first chemical species, bromate ions. A second chemical species source 104, in this case a second burette, is used to provide a selected amount of a second chemical species, in this case malonic acid. A controller 108, in this case a stopcock, one for source 106 and one for source 104, is coupled to control the addition of the first and second chemical species from the first chemical species source 106 and the second chemical species source 104, in response to an input. A sensor 114 (e.g., be a redox sensor, a pH sensor, a temperature sensor, a pressure sensor, a UV-Vis sensor or combinations thereof) can be positioned to sense changes in the reactant as the controller controls the first and second chemical species sources to add selected amounts of the respective first and second chemical species to the reactor, wherein the controller receives signals 110 (e.g., an open parenthesis) and 112 (e.g., a closed parenthesis) representing states of the reactants and correlates the states of the reactants to a result that is computed as a function of the input.

In the embodiment where the FSM is as shown in FIG. 1, the FSM, in its initial state (left-most panel) responds after an “open parenthesis” input (center panel) followed by a “close parenthesis” input (right-most panel). The Ru-(II) complex colors the FSM in its initial state (i.e., an orange hue of a given intensity) that changes to a first state (i.e., green hue of a given intensity) as the catalyst reacts and transforms into the Ru-(III) state. After the “closed parenthesis” input, the FSM initiates oscillations with a given frequency (“ƒ”) and the color oscillates between two specific intensities and hues.

In the embodiment where the FSM is a “parenthesis checker,” the tape/FSM combination checks whether a sequence of open and closed parentheses “fed” to the

Turing machine is matched. For example, the two sequences “(( ))” and “( )( )(( ))” are matched, while the sequences “(( )(” and “((( )” are not matched. In the case where the parentheses are matched, the effective procedure executed by the FSM in the chemically-operated Turing machine will give a positive (e.g., logical) answer to the first group of two inputs. In the case where the parentheses are not matched, the effective procedure executed by the FSM in the chemically-operated Turing machine will give a negative (e.g., illogical) answer to the second group of two inputs. As used herein, the term “illogical” refers to an expression that does not make sense, relative to a sequence of matched open and close parentheses.

As can be seen from FIG. 1, the procedure implemented by the chemically-operated Turing Machine of the embodiments of the present invention are general. Moreover, the chemically-operated Turing Machine of the embodiments of the present invention can be built from readily available parts and chemical substances and requires no specialized equipment. In some specialized applications, however, ad hoc reactors and feed systems (i.e., chemical species sources) may have to be designed and built for example by embedding the components necessary to carry out the B/Z reaction in polymer beads (e.g., resin beads that are loaded with the reactant) to which the “alphabet letters,” also known as the first and second chemical species, are fed, thus generating a large number of potential configurations, each of which could be programmed by the user/designer to execute some simple activity, while the ensemble leads to programmed emergent behavior, including in-phase oscillations and ensemble division, potentially leading to controlled self-replication) or within gels (e.g., agarose gel) for which chemo-mechanical coupling takes place.

In reading the result of the operation of any Turing machine, including the chemically-operated Turing Machine of the embodiments of the present invention, one needs an interface between the output of the machine and the user. For example, in standard Turing machines one interprets a “1” printed by a standard parenthesis checker mechanical Turing machine on an output tape to mean that there is a “parentheses match,” and a “0” to mean that “parentheses do not match.” This interface is referred to herein as the “nomograph” for the Turing machine of some embodiments of the present invention and it is designed to allow the user to understand the result.

EXAMPLES OF THE LOGICAL OPERATION OF EMBODIMENTS OF A CHEMICALLY OPERATED TURING MACHINE

The implementation of a computation by an embodiment of the presently described chemically operated Turing machine including a generic chemical formula (“recipe”) for its simple realization is described below. Also described, is an example of a monitoring system set-up; the logical structure of the tape; and the states and finite state machine representing the chemistry. These examples are set forth to assist in understanding the embodiments of the invention and should not, of course, be construed as specifically limiting the embodiments of the invention described and claimed herein.

The Computation

The computations carried out by the chemically-operated Turing Machine through its states and tapes, implement an “effective procedure” described in greater detail below. The examples of computations are carried out by an embodiment that consists of a parenthesis checker (see below). The effective procedure was designed to capture the special requirements of the chemical reactions described herein and enables the chemically-operated Turing machine to generate “responses” that are easily identified by direct examination of the state of the reactor after a computation, using standard chemical instrumentation such as reference electrodes or a spectrophotometer.

These responses, {<ρ>, ƒ}, obtained from the chemically-operated Turing Machine of the embodiments of the present invention correspond directly to the quantity

${\langle\rho\rangle} = \frac{\left\lbrack {{Ru}({III})} \right\rbrack}{\left\lbrack {{Ru}({II})} \right\rbrack + \left\lbrack {{Ru}({III})} \right\rbrack}$

and to the frequency f of oscillation between the two states [Ru(II)] and [Ru(III)] of a Ru-bpy that is involved in the B/Z reaction where lithium bromate represents “(” and malonic acid represents “)”; whereas <ƒ> is the value of the average oscillation amplitude.

The computation is carried out in a stirred reactor, containing 100 mL of a reactant solution with the concentrations shown in Table 1. The reactor is shielded from light to avoid interference in the calculated chemical kinetics of photosensitive reactions and is kept at a constant temperature of 25° C.±0.2° C.

TABLE 1 Initial concentrations of reagents in the 100 mL reactor H₂SO₄ 0.6M Ru(bpy)₃ ²⁺ 0.24 × 10⁻³M Ru(bpy)₃ ³⁺ 0M BrO₃ ⁻ 0M Malonic acid 0M

The drop volumes used to “write” the input tape have the characteristics shown in Table 2.

TABLE 2 “Alphabet” droplets to build the input tape BrO₃ ⁻ Malonic acid Drop volume (mL) 0.7 0.6 Concentration (M) 13.98 7.33

The highly concentrated bromate drops can be obtained from commercially available LiBrO₃. The lithium bromate and malonic acid dissolve in the reactor's solution to give a step change in the reactor's BrO³⁻ concentration of 0.1M and in the reactor's malonic acid concentration of 0.045M, respectively. These affect the extent of all reactions, hence modifying the chemical oscillation characteristics of both products as well as other specific properties of the reaction and changes the detailed properties of states of the FSM.

Using the reagents listed above in the proportions given in Tables 1 and 2, the reactor can do computations for a maximum of seven matched parentheses, as shown in FIG. 2. If more parentheses are entered, then the B/Z reaction network transitions to a steady-state regime, i.e., to a non-oscillatory regime.

A strategy to enable the computation of longer expressions without transition to a non-oscillatory regime, relies in changing the input “alphabet,” in particular increasing the relative concentration of the malonic acid in the drops with respect to the bromate drops. One should keep in mind that if the concentrations are changed, the nomograph (to be described below) has to be recalibrated. But, for a given set of concentrations, the nomograph, once available, is unique and is valid for the interpretation of any expression to be tested.

FIG. 3 shows an example in which the concentration of reagents in the drops has been changed so that when added to the reactor, the droplets dilute in the reactor's solution to give a step change in the reactor's BrO³⁻ concentration of 0.045M and in the reactor's malonic acid concentration of 0.1M, respectively. As shown, the length of the input sequences that can be computed is considerably longer than in the previous example, but the monitoring equipment would require more precision in order to detect the relative changes in oscillation mean and frequency of oscillation that are smaller than those attained with the concentrations given in Table 2.

Standard laboratory equipment can be used to monitor the metal complex oscillations and the associated oscillations in the solution color that are the response of the chemically-operated Turing Machine of the embodiments of the present invention. Typically, the redox potential and the color are monitored with the help of reference electrodes (e.g., Pt-working and a Ag-quasi reference electrode) connected to an electrometer and a spectrophotometer (e.g., monitoring absorption of 635 nm wavelength light), respectively. In embodiments where the B/Z reaction is a Ruthenium-catalyzed reaction, the chemical oscillations may be monitored using a system such as the one described below. An example of a monitoring system is shown in FIG. 4, which is a standard optical monitoring system and is available in the open literature. See, e.g., T. Amemiya et al. (2002). This system 400 comprises a diode laser 402 (e.g., one emitting 635 nm wavelength light); an optical chopper 404 to modulate the laser beam; a focusing lens 406; a neutral density filter 408; a photodiode 410 to measure the intensity of the beam 412 (broken line) after passing through the solution 414 (comprising a metal complex and other reagents) comprised in reactor 416; a color filter 418; a current pre-amplifier 420 to amplify the photocurrent signal from the photodiode 410; a two-phase lock-in amplifier 422 to further amplify the signal; and a computer 424 to receive and interpret the signal. The monitoring system 400 also comprises a temperature controller 426, connected to temperature transmitter 427, to control the temperature of the reactor 416; and temperature controller 428, connected to temperature transmitter 429, to control the temperature of the first and second chemical species sources 432 and 430, respectively. Finally, the monitoring system 400 comprises a pump controller 434 coupled to control the addition of malonic acid via conduit 440 (which is in fluid communication with the second chemical species source 430 and reactor 416) and a pump controller 436 coupled to control the addition of bromate ions via conduit 442 (which is in fluid communication with the first chemical species source 432 and reactor 416). In some embodiments, the monitoring system 400 also comprises an electrometer 438. In some embodiments, the system 400 also comprises a stirring mechanism 444 for stirring the solution 414.

The Logical Structure of the Tape and Finite State Machine Representing the Above Chemistry. Illustrated by Example 1, the Parentheses String “( ) ( )” and by Example 2, the Parentheses String “( ) ( (”

The following example illustrates how the FSM/tape combination, which make up the chemically-operated Turing machine of the embodiments of the present invention work together. These examples describe the evolution of the various states (i.e., the five states) in the chemically-operated Turing machine of the embodiments of the present invention.

The above chemistry of the chemically-operated Turing machine of the embodiments of the present invention has a structure that is equivalent to at least the following logical components and states. Note that the tapes (except for the input tape) are conceptual constructs whose role in the following is to describe the logical operation of embodiments of the chemically operated Turing machine of the embodiments of the invention.

The Logical Structure

First, a brief description is provided of the components of the logical structure for the embodiments of the present invention. See FIGS. 5A and 5B.

INPUT TAPE AND INPUT HEAD: The input tape contains a string of parentheses as supplied by the user. This is the string that the user wishes to check whether or not is logical, and if it is illogical which type of parenthesis is in excess. The head is a “read-only” device which reads each symbol on the tape in consecutive order.

LOGIC COUNTER TAPE AND ITS HEAD: This tape initially has a single “0” written on it. The head for this tape is a “read and write” device. The purpose of this component is to keep track of the number of “(” that have not yet been cancelled with a corresponding “)”. At all times the number of “(” which have not yet been cancelled is displayed on the tape in unary.

LOCATOR TAPE AND ITS HEAD: The locator tape is an infinite two-dimensional tape which serves as a filing cabinet. Said tape includes locations which can be read or written with an appropriate entry; said locations are called cells. Each cell contains a <ρ> and frequency, ƒ, pair of values associated with a certain combination of total inputs and total number of unpaired “(”. The head is a “read-only” device which moves to different cells to find the correct <ρ> and frequency, ƒ, values describing the system at a certain time. The head on this tape moves down one row for each alphabet input, and serves as a counter for the total number of inputs added up to the current time.

OUTPUT TAPE AND ITS HEAD: This tape is initially blank, and the head is a “write-only” device which writes to the tape the <ρ> and frequency, ƒ, values of the system at each time. These values are provided by the LOCATOR TAPE AND ITS HEAD, which is described below.

HEAD CONTROL: this is the central unit which controls the movement of the heads in directions specified by both the state of the chemically-operated Turing machine of the embodiments of the present invention, and the stimuli the head control receives from the various tapes.

Not all the tapes are physically implemented in the chemically-operated Turing machine of the embodiments of the present invention, where they are necessary in order to have a clear logical description of the chemically-operated Turing machine of the embodiments of the present invention.

The Operation of the Previous Logical Components.

Next, a brief description is provided of the logical operation of each of the previous components.

The INPUT TAPE AND HEAD: The tape is created by the user of the chemically-operated Turing machine of the embodiments of the present invention. It contains the string of parentheses the user wishes to determine whether or not is logical. The user must end this string with an “E” printed on the tape.

The LOGIC COUNTER TAPE AND HEAD: The tape begins with all cells blank except for one which contains a “0”. This tape and head will keep track of the number of unmatched “(” the chemically-operated Turing machine of the embodiments of the present invention encounters, while reading from the INPUT TAPE, by printing a 1 for each “(”. The total number is written in unary. This number is decreased, by replacing one of the 1's with a “0”, each time the chemically-operated Turing machine of the embodiments of the present invention reads-in an “)” that corresponds with a preceding “(”.

The OUTPUT TAPE AND HEAD: The tape is initially blank. This part of the chemically-operated Turing machine of the embodiments of the present invention will record the output after each symbol is read from the INPUT TAPE. The output is a set containing some average oscillation amplitude value, <ρ>, and a frequency of oscillation ƒ. These values are found on the LOCATOR TAPE, and are simply copied to the OUTPUT TAPE. An “X” may also be printed on the output tape; this signifies that the input string was illogical.

The LOCATOR TAPE AND HEAD acts as a filing cabinet which the chemically-operated Turing machine of the embodiments of the present invention pulls from. This is a two dimensional tape which contains pairs that specify a certain average amplitude of oscillations value, ρ, and a frequency of oscillation ƒ. These values are unique to the input read-in by the chemically-operated Turing machine of the embodiments of the present invention up to any point in time. The head of the chemically-operated Turing machine of the embodiments of the present invention moves down one cell each time an input is read-in. Additionally, the head moves right one cell if an “(” was read from the INPUT TAPE, and one cell left if an “)” was read-in.

The vertical movement of the head on the locator tape counts the total number of inputs. The left and right movements place the head over a certain set, {<ρ>, ƒ}, corresponding to a certain number of open parentheses and a certain number of closed parentheses that have been read-in by the chemically-operated Turing machine of the embodiments of the present invention.

The LOCATOR HEAD begins in the (0,0) entry of this tape, corresponding to no inputs being read from the INPUT TAPE. The column associated with this cell contains all the sets, {ρ*, ƒ*}, that correspond to an equal number of open and closed parentheses being read-in by the chemically-operated Turing machine of the embodiments of the present invention up to a certain time.

At any point in time after some input has been read-in, the number of cells away from the zero column where the head is exactly equal to the number written in unary on the LOGIC COUNTER TAPE.

The column of B's represents the cells the head will reach only if an excess of closed parentheses have been read-in, thus the string of parentheses is illogical. If the head finds a “B” in the cell it moves to, the chemically-operated Turing machine of the embodiments of the present invention halts and declares the Input “illogical”.

Blank cells on this tape represent cells that the head will never move to because their location corresponds to an impossible total counter and logic counter combination.

The States of the Finite State Machine that Make it Equivalent to the Chemistry

The five states of the chemically operated Turing machine of the embodiments of the present invention are as follows:

STATE-I (Initial): This is the first state the machine enters upon starting. It immediately filters out strings of parentheses that begin illogically (e.g., begin with “))”) by sending them to state Q2 and labeling them as “illogical”. If the string begins with “(” the chemically operated Turing machine of the embodiments of the present invention moves on to state Q1.

STATE-Q1: This state instructs the head control to move the head on the logic counter tape one cell to the right, and the head on the locator tape one cell down, and one cell to the right. In this state the machine only responds to a stimulus from the locator tape. The response to this stimulus has two parts. The first part is that the corresponding <ρ> and frequency, ƒ, pair (as provided by the locator tape) is printed on the output tape. The second part is that a “1” is printed on the logic counter tape. Note that the machine can only move to state Q3 from this state.

STATE-Q2: The machine only moves to this state if the first symbol on the Input Tape happens to be “)”. This state instructs the head control to move the head on the locator tape one cell down and one cell to the left. The head on the locator tape will then encounter a “B” written in this cell. This will cause the machine to print an “X” in the current cell on the output tape and to halt. The “X” indicates that the input string was illogical.

STATE-Q3: In this state the head control will move the heads on input and output tapes one cell to the right. In this state, the machine only responds to a stimulus from the input tape. If the stimulus is “(” the machine moves to state Q1. On the other hand, if the stimulus is “)” the machine moves on to state Q4 and the response is a “0” printed on the logic counter tape. If the logic counter tape had any 1's on the tape, this “0” will replace one of them. If the stimulus is “E” the machine moves on to state Q5.

STATE-Q4: In this state the head control moves the head of the logic counter one cell to the left, and the head on the locator tape one cell down and one cell to the left. Note that when in this state the machine only responds to a Stimulus from the locator tape. If the stimulus is a “B” the machine prints an “X” on the output tape and halts. If the stimulus is a pair of <ρ> and frequency, ƒ, from the locator tape, then the response is to print this pair on the output tape, and the machine moves on to state Q3.

STATE-Q5: In this state no heads move. The stimuli come as pairs: one part from the logic counter tape and the other from the locator tape. The part of the stimulus which comes from the locator tape may be a specific <ρ> and frequency, ƒ, pair. The part of the stimulus from the logic counter tape may either be a “1” or “0”. If “0”, the machine prints the current <ρ> and frequency, ƒ, of the system on the output tape; this signals that the string is logical and has matched parentheses. If “1”, then the machine prints an “X” on the output tape (the string is illogical because at least one “(” remained unmatched).

The Nomograph

For the example of the parenthesis matching using the chemically-operated

Turing machine of the embodiments of the present invention, the nomograph translates the machine's computations (oscillatory properties consisting of the frequency of the chemical oscillations and the average value of ρ) into one of the following four possibilities: “Yes, the Result is that the parenthesis match”; “No, the expression entered has too many open parentheses”; “No, the expression entered has too many closed parentheses”; or “No, the expression entered is illogical.”

Given a set of chemical parameters for the B/Z reaction used in the chemically-operated Turing machine of the embodiments of the present invention, the nomograph can be constructed and be used to interpret any calculation carried out by any instances of this machine. The nomograph is equivalent to a calibration curve or reference curve for the reaction and for the problem at hand solved by the chemically-operated Turing machine of the embodiments of the present invention. Therefore, given a set of reaction conditions (e.g., reactant concentration; first chemical species concentration; and second chemical species concentration), the same nomograph must be used for all the computations. However, if the reaction conditions are modified, for example, if the size or concentration of the drops used as the input alphabet is changed, then the chemical machine may need recalibration and a new nomograph may be necessary. The same applies if the underlying chemical reaction was not the B/Z system and a different chemistry was involved.

The nomograph is a ladder-curve that displays the average amplitude of oscillations only for the case of matched (or cancelling) pairs of parentheses. The nomograph is a pre-calibrated graph against which one compares the results of the computation taking place and from which the results are interpreted. The graph represents both the oscillations in p and their average value as a function of time as one adds a selected sequence of parenthesis to the FSM. More specifically, it is constructed so that it satisfies the following:

an expression with matched parentheses ends with the response printed exactly on the nomograph line in the last computed cell of the output tape;

an expression with excess open parentheses ends with the response printed above the nomograph line in the last computed cell of the output tape;

an expression with excess closed parentheses ends with the response printed below the nomograph line in the last computed cell of the output tape;

an illogical expression results in a response printed below the nomograph at the corresponding step where the expression became illogical;

for an odd number of input parentheses the response will lie either above or below the nomograph line (excess of open or closed parentheses, respectively); and

only when in configurations where there is not any excess of either open or closed parentheses does the response of the chemical system lie precisely on the nomograph curve.

Example 1 of Actual Operation: Checking the Parentheses String “( )( )”

First, the string that is to be checked is written, beforehand, on the input tape.

Additionally, an “E” is written on this tape immediately after the last parenthesis in the string. This will indicate to the machine the end of the string. See FIG. 5A.

The chemically-operated Turing machine of the embodiments of the present invention starts in state (I), where the first parenthesis is read from the input tape. The first parenthesis is open “(.” At this point, a volume of bromate ions is added. The machine moves to state (Q1). As a result the head on the logic counter tape moves right one cell and the head on the locator tape moves down one cell and right one cell.

The chemically-operated Turing machine of the embodiments of the present invention is now in state (Q1). The head on the locator tape reads, from the cell it is currently on, its average oscillation amplitude value, <ρ> and the oscillation frequency, ƒ. This causes the chemically-operated Turing machine of the embodiments of the present invention to move to state (Q3). The pair of numbers from the locator tape is printed in one cell of the output tape. Also, a “1” is printed on the logic counter tape. In addition, the heads of both the input tape and the output tape move one cell to the right.

The average oscillation amplitude value <ρ> and frequency that were written on the output tape, or equivalently, copied from the locator tape, are 1 and 0, respectively, since only one type of symbol has been read by the machine.

The chemically-operated Turing machine of the embodiments of the present invention is now in state (Q3). The next input is read from the input tape. In this case the input is a closed parenthesis “)”. See FIG. 5B. A volume of malonic acid is added.

This causes the chemically-operated Turing machine of the embodiments of the present invention to move to state (Q4). A “0” is printed on the logic counter tape, replacing the “1” that was printed previously. Then the head on the logic counter tape moves one cell to the left, and the head on the locator tape moves one cell down, and one cell to the left.

The length of time between the two inputs, “(” and “)” should be long enough so as to guarantee that the chemical reaction has reached steady state for that particular configuration.

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q4). The head on the locator tape reads, from the cell it is currently on, its average oscillation amplitude value, <ρ> and the oscillation frequency, ƒ. This causes the machine to move to state (Q3). The pair of numbers from the locator tape is printed in one cell of the output tape. The heads of both the input tape and output tape move one cell to the right.

The frequency that was written on the output tape, or equivalently, copied from the locator tape is, at this point, non-zero since both types of inputs have been read from the input tape. The average oscillation amplitude value <ρ> lies on the nomograph, since exactly one of each type of input has been added. See FIG. 6.

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q3). The next input is read from the input tape. In this case the input is “(” (i.e., a volume of bromate ions is added). This causes the chemically-operated Turing machine of the embodiments of the present invention to move to state (Q1). The head on the logic counter tape moves one cell to the right, and the head on the locator tape moves one cell down, and one cell to the right.

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q1). The head on the locator tape reads, from the cell it is currently on, its average oscillation amplitude value, <ρ> and the oscillation frequency, ƒ. This causes the chemically-operated Turing machine of the embodiments of the present invention to move to state (Q3). The pair of numbers from the locator tape is printed in the current cell of the output tape. Also, a “1” is printed on the logic counter tape. In addition, the heads of both the input tape and output tape move one cell to the right.

The frequency has increased. The average oscillation amplitude value lies above the nomograph, since at this point more “(” have been read from the input tape, than “)”.

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q3). The next input is read from the input tape. In this case, the input is a closed parenthesis “)” (i.e., a volume of malonic acid is added). This causes the machine to move to state (Q4). A “0” is printed on the logic counter tape, replacing the “1” that was printed previously. Then, the head on the logic counter tape moves one cell to the left, and the head on the locator tape moves one cell down, and one cell to the left.

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q4), the head on the locator tape reads, from the cell it is currently on, its average oscillation amplitude value, <ρ>, and the oscillation frequency ƒ. This causes the chemically-operated Turing machine of the embodiments of the present invention to move to state (Q3). The pair of numbers from the locator tape are printed in one cell of the output tape. The heads of both the input tape and output tape move one cell to the right.

Once again, the frequency has increased. The average oscillation amplitude value now lies on the nomograph, since at this point two of each type of input, that is two “(“and two ”),” have been read from the input tape.

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q3). The next input is read from the input tape. In this case the input is “E”. This signals the end of the parentheses string and causes the machine to move to state (Q5). Nothing is printed on any tapes, nor do any heads move during the transition to state (Q5).

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q5). The head on the logic counter tape reads the symbol printed in the cell it is currently on, and the head on the locator tape reads, from the cell it is currently on, its average oscillation amplitude value, <ρ> and the oscillation frequency, ƒ. The symbol read from the logic counter tape is “0” in this case. This will cause the chemically-operated Turing machine of the embodiments of the present invention to print, on the output tape, the set of numbers copied from the locator tape. Then, the chemically-operated Turing machine of the embodiments of the present invention will halt.

At this point, the frequency and average oscillation amplitude value remain unchanged since no additional input has been introduced into the chemical system.

The chemically-operated Turing machine of the embodiments of the present invention has halted. The output tape contains only a list of pairs which were copied from the locator tape. The fact that no “X” is printed on the output tape indicates that the string of parentheses on the input tape is “logical.” Also, the last two cells of the output tape contain the same pair of numbers. This shows that since no more open or closed parentheses were read from the input tape, the average oscillation amplitude value, and frequency, should remain unchanged.

Since the frequency is non-zero, and the average oscillation amplitude value lies exactly on the nomograph in the last cell and at no point in time fell below the nomograph line, this indicates that the input string of parentheses was “logical.”

The nomograph and the machine's answer for the chemically operated Turing machine of the embodiments of the present invention for the string “( )( )” are shown in FIG. 6.

Example 2 of Actual Operation: Checking the Parentheses String “( ) ( (”

First, the string that is to be checked is written, beforehand, on the input tape. Additionally, an “E” is written on this tape immediately after the last parenthesis in the string. This will indicate to chemically-operated Turing machine of the embodiments of the present invention the end of the string.

The chemically-operated Turing machine of the embodiments of the present invention starts in state (I), where the first parenthesis is read from the input tape. The first parenthesis is open “(”; therefore, the chemically-operated Turing machine of the embodiments of the present invention moves to state (Q1). As a result the head on the logic counter tape moves right one cell and the head on the locator tape moves down one cell and right one cell.

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q1). The head on the locator tape reads, from the cell it is currently on, its average oscillation amplitude value, <ρ> and the oscillation frequency, ƒ. This causes the chemically-operated Turing machine of the embodiments of the present invention to move to state (Q3). The pair of numbers from the locator tape is printed in one cell of the output tape. Also, a “1” is printed on the logic counter tape. In addition, the heads of both the input tape and the output tape move one cell to the right.

The average oscillation amplitude value <ρ> and frequency that were written on the output tape, or equivalently, copied from the locator tape, are 1 and 0, respectively, since only one type of symbol has been read by the chemically-operated Turing machine of the embodiments of the present invention.

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q3). The next input is read from the input tape. In this case the input is a closed parenthesis “)”. This causes the chemically-operated Turing machine of the embodiments of the present invention to move to state (Q4). A “0” is printed on the logic counter tape, replacing the “1” that was printed previously. Then the head on the logic counter tape moves one cell to the left, and the head on the locator tape moves one cell down, and one cell to the left.

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q4). The head on the locator tape reads, from the cell it is currently on, its average oscillation amplitude oscillation value, <ρ> and the oscillation frequency, ƒ. This causes chemically-operated Turing machine of the embodiments of the present invention to move to state (Q3). The pair of numbers from the locator tape is printed in one cell of the output tape. The heads of both the input tape and output tape move one cell to the right.

The frequency that was written on the output tape, or equivalently, copied from the locator tape is, at this point, non-zero since both types of inputs have been read from the input tape. The average oscillation amplitude value <ρ> lies on the nomograph, since exactly one of each type of input has been added. See FIG. 7.

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q3). The next input is read from the input tape. In this case the input is “(”. This causes the chemically-operated Turing machine of the embodiments of the present invention to move to state (Q1). The head on the logic counter tape moves one cell to the right, and the head on the locator tape moves one cell down, and one cell to the right.

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q1). The head on the locator tape reads, from the cell it is currently on, its average oscillation amplitude value, <ρ> and the oscillation frequency, ƒ. This causes the chemically-operated Turing machine of the embodiments of the present invention to move to state (Q3). The pair of numbers from the locator tape is printed in the current cell of the output tape. Also, a “1” is printed on the logic counter tape. In addition, the heads of both the input tape and output tape move one cell to the right.

The frequency has increased. The average oscillation amplitude value lies above the nomograph, since at this point more “(” have been read from the input tape, than “)”.

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q3). The next input is read from the input tape. In this case the input is an open parenthesis “(”. This causes the machine to move to state (Q1). The head on the logic counter tape moves one cell to the right, and the head on the locator tape moves one cell down, and one cell to the right.

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q1). The head on the locator tape reads, from the cell it is currently on, its average oscillation amplitude value, <ρ> and the oscillation frequency, ƒ. This causes the machine to move to state (Q3). The set of numbers from the locator tape are printed in one cell of the output tape. Also, a “1” is printed on the logic counter tape. In addition, the heads of both the input tape and output tape move one cell to the right.

The frequency has increased. The average oscillation amplitude value lies even further above the nomograph, since at this point two excess “(” have been read from the input tape.

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q3). The next input is read from the input tape. In this case the input is “E”. This signals the end of the parenthesis string and causes the machine to move to state (Q5). Nothing is printed on any tapes, nor do any heads move during the transition to state (Q5).

The chemically-operated Turing machine of the embodiments of the present invention is in state (Q5). The head on the logic counter tape reads the symbol printed in the cell it is currently on, and the head on the locator tape reads, from the cell it is currently on, its average oscillation amplitude value <ρ> and the oscillation frequency, ƒ. The symbol read from the logic counter tape is “1” in this case. This will cause the output tape head to print an “X” on the output tape. Then the chemically-operated Turing machine of the embodiments of the present invention will halt.

The frequency and average oscillation amplitude value remain unchanged since no additional input has been introduced into the system.

The chemically-operated Turing machine of the embodiments of the present invention has halted. The output tape contains the list of pairs copied from the locator tape. However, in the last cell printed there is an “X”. This indicates that the string of parentheses on the input tape is “illogical.” Further, the fact that a “1” was read from the logic counter tape indicates that the string of parentheses contained too many open parentheses. After processing the entire string, there remained open parentheses that had not been “cancelled” or “matched” with closed parentheses.

The average oscillation amplitude value lies above the nomograph in the last cell. This indicates an excess of “(” in the input string. Therefore, the string is “illogical.”

In the examples given above, the chemically-operated Turing machine of the embodiments of the present invention comprises a reactor that is operated in a semibatch format, with some discrete feeds of the first and second chemical species that make up the input tape added at constant time steps and no outflows from the reactor. In some embodiments, the chemically-operated Turing machine of the embodiments of the present invention can be modified such that the reactor operates as a CSTR, with continuous feed and outflow. In this case, the input string to be computed is represented as step changes of either of the two added species (i.e., the first and second chemical species) to represent “(” and “)” at fixed time intervals (longer than the residence time in the reactor).

The nomograph and the machine's answer for the chemically operated Turing machine of the embodiments of the present invention for the string “( ) ( (” are shown in FIG. 7.

Although the examples given above rely on the chemically-operated Turing machine using the oscillatory regime of the B/Z reaction, other chemically-operated Turing machines of the embodiments of the present invention are contemplated that rely on both the oscillatory and steady-state regimes of the B/Z reaction.

Finally although the examples given above, present the chemically-operated Turing machine of the embodiments of the present invention as a parenthesis checker, which is but one example, those of skill in the art could expect to construct any chemically-operated Turing machine, even universal chemical Turing machines, using the teachings of the instant disclosure.

Embodiments of the invention described and claimed herein are not to be limited in scope by the specific embodiments herein disclosed, since these embodiments are intended as illustration of several aspects of the disclosure. Any equivalent embodiments are intended to be within the scope of this disclosure. Indeed, various modifications of the embodiments in addition to those shown and described herein will become apparent to those skilled in the art from the foregoing description. Such modifications are also intended to fall within the scope of the appended claims.

All publications, including non-patent literature (e.g., scientific journal articles), patent application publications, and patents mentioned in this specification incorporated by reference as if each individual patent application publications, and patents were specifically and individually indicated to be incorporated by reference. 

1. A Turing machine based on an oscillatory chemical reaction comprising: a reactor comprising a reactant solution comprising a reactant; a first chemical species source to provide a selected amount of a first chemical species; a second chemical species source to provide a selected amount of a second chemical species; one or more controllers coupled to control the addition of the first and second chemical species from the first and second chemical species sources responsive to an input; and one or more sensors positioned to sense changes in the reactant as the controller controls the first and second chemical species sources to add selected amounts of the respective first and second chemical species to the reactor, wherein the controller receives signals corresponding to the state of the reactant and correlates the states of the reactant to a result that is computed as a function of the input.
 2. The Turing machine according to claim 1, further comprising a tape to provide the input to the controller.
 3. The Turing machine of claim 1, wherein the one or more sensors comprises a redox sensor, a pH sensor, a temperature sensor, a pressure sensor, a UV-Vis sensor or combinations thereof.
 4. The Turing machine of claim 1, wherein the first chemical species comprises an oxidizing agent and the second chemical species comprises a reducing agent.
 5. The Turing machine according to claim 4, wherein the oxidizing agent comprises bromate ions.
 6. The Turing machine according to claim 4, wherein the reducing agent comprises malonic acid.
 7. The Turing machine claim 1, wherein the reactor is a continuously stirred tank reactor.
 8. The Turing machine claim 1, wherein the sensor comprises a spectrometer adapted to periodically detect color changes in the reactant solution.
 9. The Turing machine claim 1, wherein the first chemical species source and/or the second chemical species source comprise burettes or syringe pumps.
 10. The Turing machine claim 1, wherein the reactant comprises a compound capable of attaining meta stable states or an oscillatory regime.
 11. The Turing machine claim 1, wherein the reactant comprises a transition metal complex.
 12. The Turing machine according to claim 11, wherein the the transition metal complex is a ruthenium complex, a cerium complex, an iron complex or a cobalt complex.
 13. The Turing machine according to claim 12, wherein the ruthenium complex is a tris(bipyridine)ruthenium (II) complex.
 14. The Turing machine of claim 1, wherein the input comprises a parenthesis.
 15. The Turing machine of claim 1, wherein said Turing machine is a parenthesis checker.
 16. A central processing unit of a programmable chemical computer comprising one or more Turing Machines of claim 1 or appropriate variants thereof.
 17. A programmable chemical computer comprising a central processing unit which comprises one or more Turing Machines of claim 1 or appropriate variants thereof.
 18. A method of operating a chemical Turing machine based on an oscillatory chemical reaction comprising: providing a reactor comprising a reactant solution comprising a reactant; providing an input to a controller, coupled to control the addition of a first and a second chemical species from a first chemical species source and a second chemical species source, responsive to the input; and sensing changes in the reactor as the controller controls the first and second chemical species sources to add selected amounts of the respective first and second chemical species to the reactor, wherein the controller receives signals corresponding to the states of the reactant; and correlating the states of the reactant to a result that is computed as a function of the input. 